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Introduction to pseudodifferential operators and dynamics of linear, time dependent Schrödinger equations

Course Type: 
PhD Course
Academic Year: 
March - May
20 h

Aim of the course is to introduce the basic tools of pseudodifferential calculus, and apply them to analyze the long time dynamics of linear, time dependent Schrödinger equations. A particular emphasis will be given to the problem of growth of Sobolev norms.

Course contents:

Part 1: Pseudodifferential operators

  • Symbolic calculus: composition, adjoints and quantizations
  • Continuity in $L^2$ : the Calderon-Vaillancourt theorem, Garding inequality
  • Flow generation and Egorov theorem
  • Functional calculus
  • Pseudodifferential operators on a manifold and global quantization

Part 2: Applications

  • Semiclassical normal form
  • Asymptotics of eigenvalues in special cases
  • Upper and lower bounds on the growth of Sobolev norms in linear, time depedent Schrödinger equations


[1] S. Alinhac and P. Gérard, Pseudo-differential Operators and the Nash-Moser Theorem (AMS, Graduate Studies in Mathematics, vol. 82, 2007).
[2] X. Saint Raymond, Elementary Introduction to the Theory of Pseudodifferential Operators (Studies in Advanced Mathematics, CRC Press, Boca Raton, 1991.)
[3] M. M. Wong, An Introduction to Pseudo-differential Operators (World Scientific, Singapore, 2nd ed., 1999.)
[4] D. Robert, Autour de l’Approximation Semi-Classique (Boston etc., Birkhäuser 1987).
[5] M. Taylor, Pseudo Differential Operators (Princeton Univ. Press, Princeton, N.J., 1981)
[6] A. Maspero, D. Robert: On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms. J. Funct. Anal., 273(2):721–781, 2017.
[7] D. Bambusi, B. Grebert, A. Maspero, D. Robert: Growth of Sobolev norms for abstract linear Schrödinger Equations. J. Eur. Math. Soc. (JEMS), in press.
[8] A. Maspero: Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations. Math. Res. Lett, in press 2018.
[9] A. Weinstein: Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44 (1977), no. 4, 883–892


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